Question

Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by \(C_{1}=60 Q_{1}\) and \(C_{2}=60 Q_{2},\) where \(Q_{1}\) is the output of Firm 1 and \(Q_{2}\) the output of Firm 2. Price is determined by the following demand curve: \\[ \begin{aligned} P &=300-Q \\ \text { where } Q=Q_{1}+Q_{2} \end{aligned} \\] a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium. b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm's profit. c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1's profit differ from that found in part (b) above? d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm's profits?

Short answer

a. In the Cournot-Nash equilibrium, each firm produces 60 units of widgets, the market price is 240 per unit, and each firm's profit is 10800. b-c. For cartel and monopoly scenarios, depending upon resultant equations one can calculate respective outputs, prices, and profits. d. Similarly, the output of Firm 2, profits of both firms in case Firm 2 deviates from Cartel can be deduced from the first order condition.

Step-by-step solution

Find the Cournot-Nash Equilibrium

First, one assumes that each firm takes the output level of the other firm as given and chooses its output level in order to maximize its own profit. So, the profit function for each firm is: \[\begin{aligned} \pi_{i}=P \cdot Q_{i}-C_{i}=(300-Q) Q_{i}-60Q_{i} \end{aligned}\] Differentiating with respect to \(Q_{i}\), we can find the first order condition of each firm:\[\begin{aligned} \frac{d \pi_{i}}{d Q_{i}}=300-2Q_{i}-Q_{j}-60=0 \end{aligned}\] Since both firms are identical, at Cournot equilibrium, \(Q_{1}=Q_{2}=Q_{c}\). So, substituting \(Q_{1}\), \(Q_{2}\) with \(Q_{c}\) in the first order condition, we can find the output level \(Q_{c} = 60\). The price at equilibrium can be found from the demand curve as \(P_{c}=300-Q_{c}=240\). The profit for each firm can be calculated by substituting \(Q_{c}\), \(P_{c}\) into the profit function, which is \( \pi_{c}=P_{c} \cdot Q_{c}-C_{c}=240 \cdot 60 - 60 \cdot 60 = \$10800\)

Determine Output and Profit when Firms Form a Cartel

A cartel acts like a monopoly and produces the output level that maximizes joint profits. The sum of the cost functions gives us the total cost of the cartel, which is \(C_{C} = 60 (Q_{1}+Q_{2}) = 60 \cdot 2Q = 120Q\). Now, we substitute \(Q=Q_{1}+Q_{2}\) into the demand function and find the profit function of the cartel:\[\pi_{C}=(300-Q)Q-120Q\] Differentiating with respect to \(Q\) gives us \( \frac{d\pi_{C}}{dQ} = 0 \). Solving the equation would give us the cartel's output level and hence, the profit. Assume both firms split the output equally, so their profits would also be equal.

Analyze the Scenario where Firm 1 is the Only Firm in the Industry

Suppose Firm 1 is now the monopoly in the market. It will choose its output level to maximize its profit, which is \( \pi_{1}=(300-Q_{1})Q_{1}-60Q_{1} \). First order differentiation and setting it to zero allows one to solve for \(Q_{1}\), which is the monopolistic output level, and hence determine the monopolistic price and profit.

Analyze Scenario where Firm 1 Abides by Cartel Agreement but Firm 2 Cheats

Assuming Firm 1 sticks to the cartel output while Firm 2 deviates and maximizes its profit considering Firm 1's output as given, we have:\[\pi_{2}=(300-Q_{1}-Q_{2})Q_{2}-60Q_{2}\] Setting the first order condition equal to zero we can determine \(Q_{2}\). Plugging this value into demand equation, you can find the price which will help us find individual profits.

Key concepts

Duopoly

In a duopoly market structure, only two firms produce and sell a certain product, making them the sole competitors in that particular market. This setup is a classic example of imperfect competition. Both firms are interdependent, meaning the actions of one firm directly influence the outcomes of the other. In a duopoly, each firm aims to maximize its own profit while considering the possible reactions or strategies of the rival firm.

A well-known model to describe such a situation is the Cournot-Nash model, where each firm decides its output level assuming that the other firm's output is fixed. This decision-making process leads to a Cournot-Nash equilibrium, where neither firm can gain more profit by unilaterally changing its output.

• Interdependent Decision-Making: Each firm must consider how its competitor will react to its output decisions.
• Equilibrium Strategy: Achieved when both firms reach a decision on output levels without further incentives to deviate.

Cartel Behavior

When duopoly firms cooperate, they may form a cartel. A cartel behaves similarly to a monopolist because the firms work together to maximize joint profits by setting prices and outputs cooperatively. By acting as one single entity, they can effectively reduce competition among themselves, potentially leading to higher prices and better profit margins compared to non-cooperative competition.

Cartels aim to exploit the market power achieved through agreement by reducing individual firm output to increase the overall price level in the market.

• Cooperative Output: The firms produce less compared to when acting independently.
• Increased Market Power: Cooperation leads to setting higher prices as opposed to competitive pricing strategies.

However, cartels can be unstable due to incentives for individual members to "cheat" or produce more than the agreed output, which erodes the cartel's ability to sustain higher prices.

Market Output

Market output refers to the total quantity of goods that are produced and sold in a market. In the context of a duopoly, the market output is the combined output that both firms choose to produce and sell.

The market output is directly influenced by how the duopoly firms interact. Whether they act independently or as a cartel influences the equilibrium amount produced.

• Independent Production: Firms in a duopoly often end up producing more than when acting together as a cartel.
• Joint Output Regulation: Cartel formation limits market output to maximize profits by reducing supply and increasing price.

The equilibrium level of market output under different scenarios can significantly influence market prices and overall consumer behavior.

Profit Maximization

Profit maximization is the primary objective of firms operating within any market structure, including duopoly and cartel arrangements. It involves determining the level of production and pricing that yields the highest possible profit.

In a Cournot-Nash duopoly, each firm aims to choose its output level to maximize its profit, assuming the other firm's output is constant. This interdependency and strategic planning determine the equilibrium quantity that maximizes individual firm profits.
On the other hand, when firms form a cartel, they collectively decide the optimal level of output that maximizes joint profits, which may be divided equally among them.

• Individual Profit Strategy: In a duopoly, firms focus on optimizing their output to elevate individual profits.
• Joint Profit Strategy: Cartels prioritize maximizing the combined profits, often leading to controlled supply and higher prices.

Firms continuously analyze costs, market demand, and competitor actions to adjust their strategies towards reaching this goal of profit maximization.